When the internal degree of freedom takes a finite number of values and evolves autonomously as a finite state space continuous Markov chain, then
one can view the motion as a random walk on a multi-layer system, where the layers are indexed by the internal states.
There are various studies of the asymptotic properties of such random walks including law of large numbers (asymptotic speed), invariance principle (central limit behavior), and large deviations. See e.g. [1]}, [2]}, [3]}, [4]}, [5]} and references therein.
